CN101710368A - Bayesian reliability comprehensive estimation method based on multisource degraded data - Google Patents

Abstract

The invention discloses a Bayesian reliability comprehensive estimation method based on multisource degraded data, which comprises the following steps of: 1. collecting multisource degraded data; 2. establishing a degraded statistic model and confirming a reliability function; 3. preprocessing the degraded data; 4. confirming and fusing prior distribution and posterior distribution; and 5. estimating the reliability of a product. The invention provides a method for establishing the degraded statistic model, which converts a form of the degraded data by carrying out transformation processing on the performance degrading model of the product and leads the degraded data which is not submitted to probability distribution to submit to the probability distribution, thereby leading the Bayesian method for commonly processing life data to be capable of processing the degraded data and realizing the Bayesian statistic estimation based on the degraded data; accordingly, the Bayesian method is introduced into the estimation method of an degradation accelerating test, and the estimation method of the degradation accelerating test is expanded.

Description

Bayesian reliability comprehensive estimation method based on multisource degraded data

Technical field

The present invention is a kind of at degraded data, based on the reliability comprehensive estimation method of bayesian theory, belongs to the reliability comprehensive estimation technical field.

Background technology

Usually, when the assessment reliability of products, obtain the lifetime data of product, and utilize the lifetime data that is obtained to come reliability of products is assessed by accelerated life test.But along with science and technology development, long-life, high reliability product become current main flow just gradually, so the characteristics of the object of needs assessment reliability also shift towards long-life, high reliability aspect.Because long-life, the high reliability characteristics of product, when assessing reliability of products by test, lifetime data more and more is difficult to obtain, and this has just caused certain difficulty to assessment.At this difficulty, accelerated degradation test arises at the historic moment, and a large amount of properties of product degraded data that obtains in real time during by test comes reliability of products is assessed.At present, use and quicken the Jiang Tongmin that the degeneration reliability estimation method is a BJ University of Aeronautics ﹠ Astronautics more widely, the accelerated degradation test reliability estimation method that people such as Li Xiaoyang propose based on Brownian Motion with Drift.

Owing to many-sided reasons such as product self, testing equipment, testing apparatuss, the result of single test often has certain randomness, one-sidedness, overall picture that can not the exhibiting product reliability characteristic.On the angle of assessment precision, if the information that is used to assess is many more, comprehensive more, so resulting assessment result is just accurate more, more near the actual conditions of evaluation object.In order more accurately comprehensively to assess reliability of products, just product each test figure in development process need be fully utilized, effectively utilize the data of each side and use reasonable method that it is merged, thereby obtain accurate more assessment result.Therefore, the reliability comprehensive estimation technology just becomes an important ring in the product reliability assessment.

For the method for comprehensive assessment, the many of research at present be bayes method, in the sixties in 20th century, has the people that bayes method is used for the reliability statistics analysis with regard to oneself in the world.The advantage of Bayesian reliability assessment technology is to make full use of a large amount of priori reliability informations, carries out reliability assessment in conjunction with a small amount of sample data then, thereby reaches the purpose of saving the test funds, shortening the product development cycle.These characteristics of bayes method have caused people's strong interest, and many scholars have carried out correlative study work and obtained a large amount of achievements in research in this field.To the eighties, oneself has the monograph of this respect, system and the work of at large having summed up this direction.In the last few years, scholars more paid attention to combining with reliability engineering practice, under the less situation of test figure, utilized comprehensive various prior imformations of bayes method and small sample test figure, in the hope of drawing more reasonable and believable assessment result at the scene.Pate-cornel, Coolen, scholars such as Lichtenstein and Newman are to collection, the arrangement of the expertise knowledge that exists in the engineering and rationally utilize problem to carry out correlative study.Erto, Selby and Shoukri, Elperin and Gertsbakh, scholars such as Arturo utilize prior imformation type common in the engineering, respectively the reliability assessment problem of different life-span distribution patterns are analyzed.At home, the National University of Defense technology has done many work in this respect, professor Zhang Jinhuai writes " Bayes's test analysis method " book, Zhang Shifeng studied the fusion method of multi-source empirical prior information in 2000, opened the characteristics that the Hunan reef knot closes weaponry System in Small Sample Situation analysis of experiments and assessment in 2003, with the bayes method is the research main line of System in Small Sample Situation statistical inference and blending theory, primary study be adapted to some common problems of bayes method engineering application.And aspect two of the precision of weaponry and reliability assessments, launched applied research, Duan Xiaojun had studied the Bayes's recurrence accuracy assessment method that merges based on priori in 2005, and the quantitative evaluation model of testing information in the evaluation, full army, Zhang Xiangping has studied Bayes's fusion method based on confidence level in Application in Reliability Estimation, Liu Han had studied the System in Small Sample Situation Reliability Assessment Method based on bayesian theory in 2006, theory and the practical problems of Liu Feizhen to existing in the solid propellant rocket reliability growth test, the multiple information of comprehensive utilization reliability growth test, the reliability growth test management has systematically been proposed, planning and analytical approach, Wu built each subsystem that industry utilization is formed guided missile system in 2007, the Test Information of unit, the utilization bayes method fully utilizes the test figure of each empirical prior information and different phase, and a small amount of field test data of combination is assessed the anti-warship guided missle flight reliability, Li Xinxin in 2008 by bayes method in the multistage reliability growth test, merge the historical stage data to present stage reliability level carry out comprehensive assessment, Feng Jing has studied the System in Small Sample Situation reliability information fusion method based on Bayes's Fuzzy Logic Operators.The Liu Songlin of Northwestern Polytechnical University studied with Kullback information as fusion criterion in 2006, the reliability assessment problem of cascade system under the situation with multi-source empirical prior information, people such as Shen Zheng studied under the multi-source empirical prior information fusion method in conjunction with expertise in 2008.

But mainly concentrate on based on aspect the comprehensive assessment of lifetime data at present in the research of reliability comprehensive estimation technical elements, and for the comprehensive estimation method based on degraded data, though the Zhang Yongqiang of the National University of Defense technology, Zhao's Zhao, Feng Jing, the Li Changyou of Harbin Institute of Technology, the army of defending of the Central China University of Science and Technology, people such as the Wang Jian of Tsing-Hua University are being applied to bayes method aspect the degraded data certain research is arranged, but do not have the comprehensive utilization multisource degraded data of a cover system, set up statistical model, and the concrete grammar that the degraded data under the different stress conditions is carried out comprehensive assessment.

Therefore in the reliability assessment field, carry out the increasingly extensive of reliability assessment along with using accelerated degradation test and degraded data thereof, the reliability comprehensive estimation method that proposes at accelerated degradation test and degraded data thereof is the task of top priority in present reliability assessment field.

Summary of the invention

The objective of the invention is the problem of utilizing degraded data reliability of products to be carried out comprehensive assessment in order to solve, reliability comprehensive estimation method at accelerated degradation test and degraded data thereof has been proposed, to satisfy in the present reliability assessment field, increasingly extensive use accelerated degradation test and degraded data thereof carry out the present situation of reliability assessment, and use the actual demand of comprehensive estimation method for improving the assessment precision.The present invention adopts bayes method, degraded data is carried out comprehensively, thereby realized comprehensive assessment based on the product reliability of multisource degraded data.

The present invention is a kind of Bayesian reliability comprehensive estimation method based on multisource degraded data, comprises following step:

The collection of step 1, multisource degraded data;

Step 2, set up the degeneration statistical model and determine Reliability Function;

Step 3, degraded data pre-service;

What step 4, prior distribution, posteriority distributed determines and fusion;

Step 5, assessment reliability of products;

The invention has the advantages that:

(1) the present invention proposes the method for building up of degeneration statistical model.By the properties of product degradation model is carried out conversion process, the form of degraded data is transformed, with and the degraded data of disobeying probability distribution obeyed certain probability distribution through this.Thereby make the bayes method of common processing lifetime data can handle degraded data, realized Bayesian statistics assessment based on degraded data.Therefore also just bayes method has been introduced among the appraisal procedure of accelerated degradation test, expanded the appraisal procedure of accelerated degradation test.

(2) the present invention can handle varying environment stress, the degraded data under the working stress level.By the relation between homing method and stress level the degraded data under the different stress levels is amounted to, they are folded under the same stress level, thereby improved the quantity of information that is used to assess, expanded the data area that can be used for assessing.

(3) the present invention proposes based on bayes method reliability of products to be carried out comprehensive estimation method at multisource degraded data.Will be from the degraded data in a plurality of sources, treated unification makes assessment result more comprehensively, accurately by this means that enlarged data volume under the same terms and form.And present comprehensive estimation method is mostly at lifetime data, and the comprehensive estimation method that is directed to degraded data also seldom.Therefore, method proposed by the invention has solved the problem of degraded data being carried out reliability comprehensive estimation.

Description of drawings

Fig. 1 is the process flow diagram of the method for the invention;

Fig. 2 is the fiduciary level curve map of the embodiment of the invention

Embodiment

The present invention is described in further detail below in conjunction with drawings and Examples.

The present invention is a kind of Bayesian reliability comprehensive estimation method based on multisource degraded data, and process flow diagram comprises following step as shown in Figure 1:

The collection of step 1, multisource degraded data;

1. need to determine the degradation parameter of collection;

When life-span of product and reliability are assessed, because the restriction of factors such as sample size, test period, test condition, the degraded data that only uses certain accelerated degradation test of product to be obtained is comparatively unilateral sometimes, and have a large amount of utilizable information in development stage, production phase, the operational phase of product, enlarge the data volume of assessment by these information, then can improve the precision of assessment.

Therefore, when determining to need the degradation parameter of collection, should be as the criterion with employed degradation parameter in accelerated degradation test and life-span and the reliability assessment.

2. collect product degradation information relevant in development stage, production phase, operational phase with fixed degradation parameter;

In the process of development stage of product, production phase, operational phase, have some pilot projects, test event, often there are a large amount of useful degradation information in these projects.

3. the degradation information of collecting is screened, put in order, obtain needed degraded data.Degraded data should meet the following conditions;

1) chooses the degradation information of same degradation parameter;

Be as the criterion with fixed degradation parameter, choose degradation information.

2) degradation information of obtaining is a degraded data, or degradation information can be converted into degraded data;

For form is the degradation information of degraded data, can directly use.If degradation information known certain function or the function expression that be degraded data can be converted into degraded data or obtain degraded data by method of emulation by known function, also can choose.

3) degradation mechanism of the degraded data of separate sources should be identical;

Because the degeneration of product parameters can be caused by different degradation mechanisms, for precision and the consistance that improves assessment, should be as the criterion with the degradation mechanism of product in the accelerated degradation test, chooses degraded data.

4. according to the source of degraded data, these degraded datas are divided, come from m information source, so just degraded data is divided into m group, obtain m group information source data as data;

Step 2, set up the degeneration statistical model and determine Reliability Function;

In Bayes statistical method, sample information should be obeyed population distribution.Therefore, at first degraded data should be carried out conversion processing, make it can obey certain probability distribution, obtain the form of sample information and population distribution thereof, thereby it can be applied among the bayes method.Concrete steps are as follows:

1. determine the degeneration statistical model of product degradation parameter;

Use the method that returns that the time dependent degradation model of product degradation parameter is carried out the match modeling.When specifically the product degradation process being carried out the match modeling, can set up the time dependent degeneration statistical model of degradation parameter by following three kinds of methods as required:

1). suppose that the parameter in the time dependent definite function model of degradation parameter obeys certain distribution;

2). to the time dependent definite function of degradation parameter, add one and do not change the margin of error of obeying certain distribution in time;

3). to the time dependent definite function of degradation parameter, add stochastic process and describe;

2. set up the degeneration statistical model and determine Reliability Function;

Different situations when setting up degradation model are carried out conversion to degradation model, and degraded data are handled, and obtain applying to distribution and data mode in the bayes method; And, set up the degeneration statistical model of product and determine Reliability Function based on this in conjunction with inefficacy thresholding l and failure criteria;

Three kinds of modeling methods at mentioning in the step 1 have proposed its processing scheme respectively:

1) supposes that the parameter in the time dependent definite function model of degradation parameter obeys certain distribution;

By certain distribution that parameter in the function is obeyed, other parameter in the function and variable are brought in this distribution, thereby set up a new distribution.And the degraded data behind the variation used as new data, set up the degeneration statistical model, obtain Reliability Function R (t);

Illustrate: suppose that the time dependent definite function model of certain product degradation parameter is:

y(t)＝β ₀+β ₁t??????????????(1)

Wherein y (t) is the product degradation parameter, β ₀, β ₁Be parameter item, t is the time; If β ₁N (θ ₁, σ ₁ ²), according to the character of normal distribution, can obtain:

y(t)□N(β ₀+θ ₁t，σ ₁ ²t ²)???????????(2)

And then obtain:

Will

As new data mode, formula (3) distributes as it, thereby has determined the form of distribution and data, has set up the degeneration statistical model;

If establish the failure threshold that l is a parameter, promptly establish y (t)-l＜0 o'clock product failure, so just can obtain the Reliability Function of product by the formula of deriving above:

$R\left(t\right)=P(y\left(t\right)-l>0)=\mathrm{\Φ}\left(\frac{{\mathrm{\β}}_0+{\mathrm{\θ}}_{1}t-l}{\sqrt{{\mathrm{\σ}}_1^2{t}^2}}\right)---\left(4\right)$

2), add one and do not change the margin of error of obeying certain distribution in time to the time dependent definite function of degradation parameter;

By certain distribution that the margin of error in the function is obeyed, parameter in the function and variable are brought in this distribution, thereby set up a new distribution.And the degraded data behind the variation used as new data, set up the degeneration statistical model, obtain Reliability Function R (t);

Now illustrate:

Suppose that the time dependent definite function model of certain product degradation parameter is:

y(t)＝Ct+ε?????????????????????????????????(5)

Wherein y (t) is the product degradation parameter, and C is a constant, ε N (θ, σ ²), t is the time; Then can obtain according to the character of normal distribution

y(t)□N(Ct+θ，σ ²)?????????????????????????(6)

And then obtain

y(Δt)□N(CΔt，0)??????????????????????????(7)

As new data mode, formula (7) distributes as it with y (Δ t), thereby has determined the form of distribution and data, has set up the degeneration statistical model;

If establish the failure threshold that l is a parameter, promptly establish y (t)-l＜0 o'clock product failure, so just can obtain the Reliability Function of product by the formula of deriving above:

$R\left(t\right)=P(y\left(t\right)-l>0)=\mathrm{\Φ}\left(\frac{\mathrm{Ct}+\mathrm{\θ}-l}{\sqrt{{\mathrm{\σ}}^2}}\right)---\left(8\right)$

3), add stochastic process and describe to the time dependent definite function of degradation parameter;

For adding the time dependent definite function of performance degradation parameter that stochastic process is described, according to the character of stochastic process function is handled, set up the degeneration statistical model, obtain Reliability Function R (t);

As for the Brownian Motion with Drift model:

Y(t)＝σB(t)+g(t，s)·t+y ₀??????????????????????????????(9)

Wherein Y (t) is the degenerative process of product parameters; B (t) is 0 for average, variance be time t standard Brownian movement B (t) N (0, t); σ is a coefficient of diffusion, does not change with stress and time, is constant; (t s) is coefficient of deviation to g; y ₀Initial value for properties of product;

Therefore by the character of Brownian Motion with Drift as can be known, it is that (variance is σ to g for t, s) Δ t that the degeneration increment Delta Y of unit interval Δ t obeys average ²The normal distribution of Δ t, promptly

ΔY□N(g(t，s)·Δt，σ ²Δt)???????????????????????????(10)

As new data mode, formula (10) distributes as it with Δ Y, thereby has determined the form of distribution and data, has set up the degeneration statistical model;

If establish the failure threshold that l is a parameter, promptly establish Y (t)-l＜0 o'clock product failure; Can utilize the Brownian Motion with Drift Reliability Model that Reliability Function is found the solution:

$R\left(t\right)=\mathrm{\&Phi;}\left[\frac{{l-y}_0-g(t,s)t}{\mathrm{\&sigma;}\sqrt{t}}\right]-\exp \left(\frac{2g(t,s)(l-y_0)}{{\mathrm{\&sigma;}}^2}\right)\mathrm{\&Phi;}\left[\frac{l-{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+g(t,s)t}{\mathrm{\&sigma;}\sqrt{t}}\right]---\left(11\right)$

Step 3, degraded data pre-service;

Because the source of the degraded data of need handling is not quite similar, and has a certain distance between data, they need be carried out certain processing after, just can be applicable to assess.Concrete steps are as follows:

1. resulting degraded data is converted into the form of obeying population distribution;

According to distribution that obtains in the step 2 and new data mode, according to the new data form formula that obtains degraded data is carried out conversion, make it obey resulting distribution;

2. definite stress is amounted to model;

Usually, the degenerative process of product is subject to as temperature, vibration, humidity, the influence of electric stress equal stress.Under different stress levels, the process of degeneration is different, and also therefore and different the meaning of degraded data representative will fully utilize multisource degraded data, and primary what solve is exactly the unification stress level, and they are folded to the condition of the reality of needs assessment.

Each stress all has its acceleration model or reduced factor separately, determines acceleration model or reduced factor according to the concrete condition and the residing stress thereof of degradation parameter, amounts to model as stress.

It is as follows now to enumerate acceleration model:

1) Allan Nice model

For temperature, use Allan Nice (Arrhenius) model usually, its form is as follows:

ξ＝Ae ^E/KT

Wherein ξ is certain life characteristics, and A is a positive constant, and E is the activation energy relevant with material, and K is a Boltzmann constant, and T is an absolute temperature.

2) contrary power law model

For electric stress, use contrary power law model usually, its form is as follows:

ξ＝Av ^-c??????????????????????????????(12)

Wherein ξ is certain life characteristics, and A is a positive constant, and c is the positive constant relevant with activation energy, and v is a stress.

3) broad sense Ai Lin model

During for temperature and voltage conduct acceleration simultaneously stress, use broad sense Ai Lin model usually, its form is as follows:

$\mathrm{\&xi;}=\frac{A}{T}\exp \left\{\frac{B}{\mathrm{kT}}\right\}\exp \left\{V(c+\frac{D}{\mathrm{kT}})\right\}---\left(13\right)$

Wherein ξ is certain life characteristics, and A, B, C, D are undetermined constant, and k is a Boltzmann constant, and c is the positive constant relevant with activation energy, and T is an absolute temperature, and V is a voltage.

4) Peck model

When stress is quickened in conduct simultaneously for temperature and humidity, use the Peck model usually, its concrete form is

Wherein ξ is certain life characteristics, Be relative humidity, Ea is an activation energy, and k is a Boltzmann constant, and T is an absolute temperature, and m and A are constant.

5) other

For some comparatively typical product and material, for its residing stresses typical, the acceleration model of himself is arranged all, in a lot of standards and document, can find.

Simultaneously, for some products, the relation (reduced factor) of its parameter between each stress level all to provide or to make mutually deserved regulation, in this case, can not passed through acceleration model, and directly use reduced factor different stress levels amounted to.

Therefore, according to each stress level in each source degraded data and the accelerated degradation test,, determine mutually deserved acceleration model or reduced factor in conjunction with the characteristics of product self.

3. the degraded data under the different stress levels is folded under the same level;

1. if it is acceleration model G (s) that the middle stress of above-mentioned steps (2) is amounted to model,, uses the method match of regression fit and also set up a certain eigenwert z relevant in the degenerative process with the time at first according to the degraded data under the different stress level s _sAnd the relational expression between the stress level s:

z _s＝G(s)?????????????????????????????????????????????????(15)

Can obtain a certain eigenwert z relevant under the different stress level s by formula (15) with the time _sValue;

Secondly, with stress s _iDegraded data y under the level _iBe folded to required stress level s _iUnder degraded data y _i, if can set up degraded data y and a certain eigenwert z relevant with the time _sRelation:

y＝f(z _s)?????????????????????????????????????????????????(16)

Wherein, formula (16) can pass through degradation model and a certain eigenwert z relevant with the time _sConcrete meaning determine; So as can be known by formula (15), (16):

$y_i=\frac{f\left(G\left(s_i\right)\right)}{f\left(G\left(s_j\right)\right)}\&CenterDot;y_j---\left(17\right)$

Thereby finished amounting between the degraded data under the different stress levels;

2. be reduced factor k if the middle stress of above-mentioned steps (2) is amounted to model _Ij, by the reduced factor k between given stress level _IjWith stress level s _iUnder a certain eigenwert z _SiBe folded to stress level s _jUnder a certain eigenwert z _Sj:

z _si＝k _ij·z _sj????????????????????????????????????????????(18)

Finished amounting to of degraded data between different stress levels by formula (16) then;

4. the form of each information source data is unified;

If can not set up such degraded data y of formula (16) and a certain eigenwert z relevant in the step 3 with the time _sRelation, then utilize the time interval Δ t between each degraded data to set up it and a certain eigenwert z relevant with the time _sRelation, then have:

Δt＝f(z _s)????????????????????????????????????????????????(19)

Obtain by formula (15), (19):

$\mathrm{\&Delta;}{t}_i=\frac{f\left(G\left(s_i\right)\right)}{f\left(G\left(s_j\right)\right)}\&CenterDot;\mathrm{\&Delta;}{t}_j---\left(20\right)$

Along with the change of the time interval Δ t between degraded data, each degraded data has been represented the parameter degradation values in the different time, need unify the degraded data form, and concrete grammar is:

At first, obtain each degraded data time interval Δ t by formula (20) _i, obtain their lowest common multiple Δ t _G, obtain Δ t _GWith Δ t _iMultiple concern p _i=Δ t _G/ Δ t _i

Then, with every p of each degraded data _iIndividual data point merges, with p _iThe amount of degradation of individual degraded data merges to a point, and then the time interval of all degraded datas all equates, the degraded data unity of form;

5. consistency check;

Each information source data that step 4 is obtained as the prior imformation in the bayes method, will speed up in the degradation experiment degraded data through step 4 as the sample information in the bayes method, and the distribution that step 2 is obtained is as population distribution; Because selected prior imformation comes from a plurality of different information sources.So the prior imformation and the sample information in each source to be carried out consistency check, whether compatible by checking two parents, whether available to determine prior imformation.

Whether the present invention uses the rank test method to check prior imformation compatible with sample information.Concrete grammar is:

$Y_i=({\mathrm{<figure-callout\; id="1"\; label="Y\; i"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">Y</figure-callout>}}_i,\&CenterDot;\&CenterDot;\&CenterDot;,Y_{{\mathrm{in}}_1})$ Be i group prior imformation, $X=({\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">X</figure-callout>}}_1,\&CenterDot;\&CenterDot;\&CenterDot;,X_{n_2})$ Be sample information, introduce and compete to select hypothesis mutually:

H ₀: X and Y _iBelong to same overall; H ₁: X and Y _iDo not belong to same overall.

If X _iWith Y _iBy check, think that two increments do not belong to same overall; Otherwise it is same overall that two increments belong to, and this group prior imformation is removed.

What step 4, prior distribution, posteriority distributed determines and fusion;

1. according to prior imformation, population distribution, the actual conditions of sample information are selected definite method of prior distribution;

According to prior imformation, population distribution, the actual conditions of sample information are determined prior distribution.The definite methods of bayesian prior distribution such as prior distribution method, empirical Bayes method of can determining by no The non-information prior distribution method, conjugation prior distribution method, maximum entropy prior distribution method, bootstrap and stochastic weighted method are determined prior distribution.

2. determine the prior distribution of each information source;

For the prior imformation of m information source, need organize prior imformation according to each and use identical prior distribution to determine definite separately its prior distribution π of method _i(θ), and π _i(θ) obey identical prior distribution family.

3. robustness check;

Adopt the method for robustness check to judge whether determined prior distribution meets the requirements.If known prior distribution π _i(θ), population distribution f (x| θ) and sample information x, the formula of Marginal density function, is so:

$m_i\left(x\right)=\underset{\mathrm{\&Theta;}}{\&Integral;}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;}---\left(21\right)$

By the m that tries to achieve _i(x) value judges whether determined prior distribution is sane, sets an acceptable value cv as required, if m _i(x)＜cv then needs to use other method to redefine prior distribution;

4. by sample information,, determine that the posteriority of each information source distributes in conjunction with the prior distribution of each information source;

By prior distribution π _i(θ), population distribution f (x| θ) and sample information x obtain the posteriority distribution π of θ _i(θ | x), the posteriority distribution formula is:

${\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right|x)=\frac{{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})}{\underset{\mathrm{\&Theta;}}{\&Integral;}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;}}---\left(22\right)$

5. the posteriority of finding the solution and obtaining after each information source merges by weighting coefficient distributes;

A plurality of prior distributions are weighted fusion, it is fused into a prior distribution.Here use correlation coefficient process in the weighting fusion method as the method that merges prior distribution.

Known population distribution f (x| θ) obtains prior distribution π respectively by m information source ₁(θ), π ₂(θ) ... π _m(θ), the prior distribution after merging so is:

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)---\left(23\right)$

ε wherein _iBe weighting coefficient:

$\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i=1---\left(24\right)$

Adopt correlation coefficient process to find the solution ε _i

This moment, posteriority was distributed as:

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right|x)=\frac{\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})}{\underset{\mathrm{\&Theta;}}{\&Integral;}\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;}}---\left(25\right)$

So the posteriority after merging is distributed as

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right|x)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&epsiv;}}_{i}m\left(x\right|{\mathrm{\&pi;}}_i)}{m\left(x\right|\mathrm{\&pi;})}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right|x)---\left(26\right)$

Wherein, $m\left(x\right|{\mathrm{\&pi;}}_i)=\underset{\mathrm{\&Theta;}}{\&Integral;}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;},$ M (x| π)=∑ m (x| π _i); Order ${\mathrm{\&xi;}}_i=\frac{{\mathrm{\&epsiv;}}_{i}m\left(x\right|{\mathrm{\&pi;}}_i)}{m\left(x\right|\mathrm{\&pi;})},$ So

$\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i=1---\left(27\right)$

Be so

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right|x)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right|x)---\left(28\right)$

Secondly, to π _i(θ) get expectation u _i, to π _i(θ | x) get expectation u ' _i, variances sigma _i ², to π (θ | x) get expectation $u^{\&prime;}=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i{u}_i^{\&prime;};$ By the emulation sampling, obtain m+1 group u _i, u ' _i, u ' estimated value, and σ ₁ ², σ ₂ ²Estimated value; Bring data into system of equations:

$\left\{\begin{array}{c}{u}_i^{\&prime;}=b_{0i}^{\&prime;}+b_{1i}^{\&prime;}{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1+b_{2i}^{\&prime;}{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_2+\&CenterDot;\&CenterDot;\&CenterDot;b_{\mathrm{mi}}^{\&prime;}{u}_m\\ u^{\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+b_1{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1+b_2{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_2+\&CenterDot;\&CenterDot;\&CenterDot;b_m{u}_m\\ r_i=\frac{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{b}_{\mathrm{ji}}^{\&prime;}{b}_j{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}{\sqrt{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{b}_{\mathrm{ji}}^{\&prime;2}{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}\sqrt{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="2"\; label="b\; j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">b</figure-callout>}}_j^{}{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}}\\ {\mathrm{\&epsiv;}}_i=r_i/\left(\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{r}_i\right)\end{array}\right.$

Obtain ε thereby find the solution _iAnd ξ _iValue; And then the posteriority distribution π of formula (28) after obtaining merging (θ | x);

Step 5, assessment reliability of products;

(1) by the posteriority distribution π after merging (θ | x),, obtain the assessed value of the parameter θ of population distribution in conjunction with the Bayesian Estimation method

Wherein:

$\widehat{\mathrm{\&theta;}}=E\left(\mathrm{\&theta;}\right|x)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i{E}_i\left(\mathrm{\&theta;}\right|x)---\left(29\right)$

(2) will

Among the Reliability Function R of substitution step 2 (t), obtain the fiduciary level value of product at moment t; Ask constantly the interval of the reliability R (t) of t to estimate as needs, by posteriority distribution π (θ | x) interval that obtains parameter θ is estimated, estimates in the interval of the reliability R (t) of moment t thereby obtain product.

Embodiment:

With certain type photovoltaic is example, by assessment to its stepstress accelerated degradation test, and the concrete Bayesian reliability comprehensive estimation method of introducing based on degraded data.

The collection of step 1, multisource degraded data;

1. need to determine the performance degradation parameter of collection;

By analyze finding that luminous power degradation failure is the main failure cause of this product, in various tests also be with this as the main performance index of product, so selective light power is as the performance parameter that should collect.

2. collect product degradation information relevant in development stage, production phase, operational phase with fixed degradation parameter;

Analyze through collecting, think that the performance parameter degraded data of this product in quickening the degeneration trial test has stronger value, various conditions and temperature stepstress accelerated degradation test are approaching, and sample size is more, its degraded data can be used as replenishing of stepstress accelerated degradation test data, can be applied to comprehensive assessment.Therefore, select data in the acceleration degeneration trial test of this product as the source of prior imformation.

With the data of stepstress accelerated degradation test source as sample information.

3. the degradation information of collecting is screened, put in order, obtain needed degraded data;

Select 2 products in the temperature stepstress acceleration degeneration trial test in the trial test, it is numbered 1,2; Select 2 products in the temperature constant stress acceleration degeneration trial test, it is numbered 3,4; With the luminous power degraded data of these 4 products as prior imformation.It is as shown in table 1 to choose the residing stress of each product

Each product stress information of table 1

With the data of temperature stepstress accelerated degradation test as sample information.Its test temperature stress is respectively 60 ℃, 80 ℃, 100 ℃ and 110 ℃.Select wherein 4 product data, it is numbered 5,6,7 and 8.

4. according to the source of degraded data, these degraded datas are divided, come from m information source, so just degraded data is divided into m group as data;

Because it is not strong to be numbered 1,2 the consistance that product showed in trial test, belong to same parent and be numbered 3,4 product.Therefore they are divided into three information sources here and handle, information source 1 is for to be numbered 1 product, and information source 2 is for to be numbered 2 product, and remaining product then is classified as information source 3.With these 3 information sources as prior imformation.

To be numbered 5,6,7 and 8 degraded data as sample information.

Step 2, set up the degeneration statistical model and determine Reliability Function

1. determine the degradation model of product degradation parameter;

Select Brownian Motion with Drift that the degenerative process of particular product performance parameters is described.Known drift Brownian movement degradation model is:

Y(t)＝σB(t)+d(s)·t+y ₀

Wherein Y (t) is the performance degradation process of product; B (t) is 0 for average, and variance is the standard Brownian movement of time t, and B (t) N (0, t); σ is a coefficient of diffusion, does not change with stress and time, is constant; D (s) is a coefficient of deviation; y ₀Initial value for properties of product.

2. set up the degeneration statistical model and determine Reliability Function;

Because Brownian Motion with Drift belongs to a kind of stochastic process, therefore, can handle function according to the character of stochastic process for adding the time dependent definite function of performance degradation parameter that stochastic process is described.

By the character of Brownian Motion with Drift as can be known, it is d (s) Δ t that the degeneration increment Delta Y of unit interval Δ t obeys average, and variance is σ ²The normal distribution of Δ t, promptly

ΔY□N(d(s)·Δt，σ ²Δt)

Therefore with normal distribution N (d (s) Δ t, σ ²Δ t) as population distribution, with degeneration increment Delta Y as sample information.

If establish l is the inefficacy thresholding of performance parameter, and the degeneration of luminous power is successively decreased, i.e. product failure when Y (t)-l＜0.Can utilize the Brownian Motion with Drift Reliability Model that Reliability Function is found the solution:

$R\left(t\right)=\mathrm{\&Phi;}\left[\frac{l-y_0-d\left(s\right)t}{\mathrm{\&sigma;}\sqrt{t}}\right]-\exp \left(\frac{2d\left(s\right)(l-y_0)}{{\mathrm{\&sigma;}}^2}\right)\mathrm{\&Phi;}[-\frac{l-{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+d\left(s\right)t}{\mathrm{\&sigma;}\sqrt{t}}]$

The pre-service of step 3, degraded data;

1. resulting degraded data is converted into new data mode;

Because it is d (s) Δ t that the degeneration increment Delta Y of degraded data obeys average, variance is σ ²The normal distribution of Δ t.Therefore all degraded datas are converted into the form of degeneration increment Delta Y.

2. definite stress is amounted to model;

Because product duty in step stress test and trial test is identical, suffered environmental stress is a temperature, selects the Arrhenius model as the temperature acceleration model, and promptly stress is amounted to model.The stress s of this moment is a temperature T, order

d(T)＝Aexp(-B/T)。

Wherein: T is an absolute temperature, K

B=Ea/k, k are Boltzmann constants 8.6171 * 10 ^-5EV/K, Ea are activation energys, eV

A is a constant

3. the degraded data under the different stress levels is folded under the same level;

Prior imformation is carried out preliminary processing and amounted to, the test figure under the different temperatures is folded to same temperature levels, for ensuing work is prepared.Here use the method for linear fit to obtain relation between temperature and the degradation trend, thereby obtain degradation ratio and its relation between the degradation ratio under 25 ℃ of three information sources under its place temperature.

It is as follows to set up equation of linear regression

E(Y(t))＝d(T)·t+y ₀

E(Y(t))＝Aexp(-B/T)·t+y ₀

Promptly

E(ΔY(t))＝Aexp(-B/T)·Δt

And then obtain linear regression model (LRM)

ln(E(ΔY(t)))＝-B/T+ln(AΔt)

4. the form of each information source data is unified;

Can obtain the parameter of acceleration model by match, use the method for amounting to detection time to realize degeneration increment amounting under each stress here to 25 ℃ of following degeneration increments.Amounting to formula detection time is

Wherein t is detection time, t _kFor with stress s _kBe folded to the detection time after 25 ℃.

For prior imformation, former detection time, interval was t=6 minute, and become 120 minutes the detection time of amounting to back information source 1 and 2 at interval; Become 240 minutes the detection time of No. 3 products at interval in the information source 3, and become 300 minutes 4 detection time at interval.

In like manner sample information is amounted to, use the method for linear regression, obtain the parameter of acceleration model, amounting to detection time.For temperature stepstress accelerated degradation test, become its detection time at each temperature: become 100 minutes 60 ℃ detection time at interval, become 300 minutes 80 ℃ detection time at interval, and become 800 minutes 100 ℃ detection time at interval, and become 1200 minutes 110 ℃ detection time at interval.

Because 2400 be about 120,240,300,100,800,1200 lowest common multiple, therefore select 2400 minutes as its detection time at interval.Degeneration increment in per 2400 minutes is merged, make a plurality of data wherein become data, make all data all with the formal representation of the degeneration increment in 2400, thereby each information has been carried out unified amounting to.Concrete combination situation is as shown in table 2.

The unification of table 2 data is amounted to

Thereby the data structure of having unified priori data and sample data is for next step data processing and assessment are got ready.

5. consistency check

Use the rank test method of Wilcoxon-Mann-Whitney to check prior imformation whether compatible with sample information.

By the rank test program ranksum among the MATLAB each group data is tested, the result of check is, information source 1 is compatible with sample information with information source 3, and information source 2 is incompatible with sample information.Therefore information source 2 is removed, selected information source 1, information source 3 enters next step assessment.

What step 4, prior distribution, posteriority distributed determines and fusion

1. according to prior imformation, overall information, the actual conditions of sample information are selected definite method of prior distribution;

Because it is more convenient that the conjugation prior distribution calculates, some parameters of posteriority distribution simultaneously can obtain good explanation, and this assessment selects for use conjugation prior distribution method to determine the prior distribution of product.Wherein the conjugation prior distribution is defined as: for a certain distribution, if by sampling information calculate posterior density function with prior distribution π (θ) identical functional form is arranged, claim that then π (θ) is (nature) conjugation prior distribution of θ.

2. determine the prior distribution of each information source;

Known Δ Y N (d (s) Δ t, σ ²Δ t), so according to conjugation prior distribution theory, the product Normal Distribution in the coefficient of deviation and the time interval as shown in Table 3, the product in the coefficient of diffusion and the time interval is obeyed the distribution of falling the gamma, i.e. d (s) t N (μ ₀, τ ²), σ ²t IG (b, a).

Table 3 conjugation prior distribution

Population distribution	Parameter	The conjugation prior distribution
			Normal distribution (variance is known)	Average	Normal distribution
Normal distribution (average is known)	Variance	Fall Gamma distribution

Its probability density function concrete form is:

By priori data Y=(y ₁, y ₂... y _m) determine prior distribution, so the characteristics of binding data are selected formula for use

$a=\frac{1}{2}\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{(y_i-\stackrel{\&OverBar;}{y})}^2$

$b=\frac{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">m</figure-callout>}}{2}$

${\mathrm{\&mu;}}_0=\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{y}_i$

${\mathrm{\&tau;}}^2=\frac{1}{m}\&CenterDot;\frac{1}{m-1}\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{(y_i-{\mathrm{\&mu;}}_0)}^2$

Determine the value of each super parameter of prior distribution, thereby determined prior distribution.The value of super parameter is as shown in table 4

The value of the super parameter of table 4 prior distribution

3. robustness check;

Prior distribution σ for coefficient of diffusion and time product ²(b a), judges its robustness by its Marginal density function, value here to t IG.

If x=is (x ₁, x ₂... x _n) be sample information, X N (θ, σ ²).Its Marginal density function, is so

Can check the robustness of prior distribution by the Marginal density function, that obtains.Here the average substitution of y is tested.The assay of information source 1,3 is 0.98,0.95.Set acceptable value cv=0.6, presentation of results Marginal density function, value is greater than acceptable value, and prior distribution has passed through check.

Prior distribution d (s) t N (μ for coefficient of deviation and time interval product ₀, τ ²), the concrete grammar of its robustness check is as follows.

If X=is (x ₁, x ₂... x _n) be sample information, X N (θ, σ ²).At first calculate m (x| π), x is the average of increment here, and its distribution density function is N (μ ₀, σ ²/ n+ τ ²).If center variable quantity Z N (0,1) can obtain so

$Z=\frac{\stackrel{\&OverBar;}{x}-{\mathrm{\&mu;}}_0}{{({\mathrm{\&sigma;}}^2/n+{\mathrm{\&tau;}}^2)}^{0.5}}$

By the Z test of normal distribution, just can check the robustness of prior distribution.The result of check passes through check for prior distribution, and the probability that passes through is respectively 0.83,0.75.

4. by sample information,, determine that the posteriority of each information source distributes in conjunction with the prior distribution of each information source;

When finding the solution the posteriority distribution, because Δ X N (d (s) Δ t, σ ²Δ t) in, d (s) Δ t, σ ²Δ t is all unknown, therefore at first determines σ by the normal state-Gamma distribution of falling ²If the super parameter that Δ t posteriority distributes is Y=(y ₁, y ₂... y _m) be prior imformation, X=(x ₁, x ₂... x _n) be posterior information, then:

$\mathrm{\&alpha;}={\mathrm{\&alpha;}}_0+\frac{n}{2}\left(\frac{1}{n-1}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(x_i-\stackrel{\&OverBar;}{x})}^2\right)+\frac{n}{2}\frac{{(\stackrel{\&OverBar;}{x}-\stackrel{\&OverBar;}{y})}^2}{n/\mathrm{<figure-callout\; id="1"\; label="m\; +"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">m</figure-callout>}+1}$

$\mathrm{\&beta;}={\mathrm{\&beta;}}_0+\frac{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">n</figure-callout>}}{2}$

By super parameter alpha ₁, β ₁Value can obtain σ ²The posteriority distribution of Δ t, expectation value, variance yields:

By the σ that obtains ²The assessed value of Δ t can draw the expression formula that d (s) Δ t posteriority distributes in conjunction with the character of conjugation prior distribution and is

Wherein,

The super parameter of resulting posteriority distribution is as shown in table 5

The value of the super parameter of table 5 posteriority distribution

5. the posteriority of finding the solution and obtaining after each information source merges by weighting coefficient distributes;

Find the solution two corresponding weighting coefficients of parameter respectively by the correlation coefficient process in the weighting fusion method.

For σ ²Δ t is at first by resulting prior distribution π ₁(σ ²Δ t), π ₂(σ ²Δ t) and posteriority distribution π ₁(σ ²Δ t|x), π ₂(σ ²Δ t|x) carries out the emulation sampling, obtain the emulation assessed value μ of its expectation value ₁, μ ₂, μ ' ₁, μ ' ₂And posteriority distribution variance emulation assessed value σ ₁ ², σ ₂ ²Each 3 groups, and Marginal density function, value m (the x| π that distributes by posteriority ₁), m (x| π ₂) obtain:

${\mathrm{\&xi;}}_1=\frac{{\mathrm{\&epsiv;}}_{1}m\left(x\right|{\mathrm{\&pi;}}_1)}{m\left(x\right|{\mathrm{\&pi;}}_1)+m\left(x\right|{\mathrm{\&pi;}}_2)}$

${\mathrm{\&xi;}}_2=\frac{{\mathrm{\&epsiv;}}_{2}m\left(x\right|{\mathrm{\&pi;}}_2)}{m\left(x\right|{\mathrm{\&pi;}}_1)+m\left(x\right|{\mathrm{\&pi;}}_2)}$

Thereby pass through formula:

π(σ ²Δt|x)＝ξ ₁π ₁(σ ²Δt|x)+ξ ₂π ₂(σ ²Δt|x)

To π (σ ²Δ t|x) carries out the emulation sampling, obtain 3 groups and contain unknown number ε ₁, ε ₂Expectation emulation assessed value μ '.With resulting 3 groups of μ ', μ ₁, μ ₂, μ ' ₁, μ ' ₂, σ ₁ ², σ ₂ ²Value substitution system of equations:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1^{\&prime;}=b_{01}^{\&prime;}+b_{11}^{\&prime;}{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1+b_{21}^{\&prime;}{u}_2\\ {\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_2^{\&prime;}=b_{02}^{\&prime;}+b_{12}^{\&prime;}{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1+b_{22}^{\&prime;}{u}_2\\ u^{\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+b_1{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1+b_2{u}_2\\ r_i=\frac{\underset{j=1}{\overset{2}{\mathrm{\&Sigma;}}}{b}_{\mathrm{ji}}^{\&prime;}{b}_j{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}{\sqrt{\underset{j=1}{\overset{2}{\mathrm{\&Sigma;}}}{b}_{\mathrm{ji}}^{\&prime;2}{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}\sqrt{\underset{j=1}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="2"\; label="b\; j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">b</figure-callout>}}_j^{}{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}}\\ {\mathrm{\&epsiv;}}_i=r_i/\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{r}_i\right)\end{array}\right.$

Use MATLAB solving equation group, thereby obtain σ ²The weighting coefficient that the prior distribution of Δ t and posteriority distribute, information source 1 is 0.1685, information source 3 is 0.8315.In like manner can obtain the prior distribution of d (s) Δ t and the weighting coefficient that posteriority distributes, information source 1 is 0.8866, and information source 3 is 0.1134.

Therefore the posteriority after can obtaining merging distributes

π(d(s)·□t|x)＝0.8866π ₁(d(s)·□t|x)+0.1134π ₂(d(s)·□t|x)

π(σ ²·□t|x)＝0.1685π ₁(σ ²·□t|x)+0.8315π ₂(σ ²·□t|x))

Step 5, assessment reliability of products

Posteriority distribution π after 1. passing through to merge (θ | x),, obtain the assessed value of the parameter θ of population distribution in conjunction with the Bayesian Estimation method

Distribute by the posteriority after merging, the character of being expected by the posteriority function can obtain

E(d(s)·□t|x)＝0.8866E ₁(d(s)·□t|x)+0.1134E ₂(d(s)·□t|x)

E(σ ²·□t|x)＝0.1685E ₁(σ ²·□t|x)+0.8315E ₂(σ ²·□t|x)

And

E _i(d(s)·□t|x)＝u _1i

So

Because 2400 minutes is 40 hours, gets t=40 hour here, thus the coefficient of deviation and the coefficient of diffusion of the product degradation when obtaining temperature and being 25 ℃, $\hat{d}\left(s\right)=1.07e-4,$ $\widehat{{\mathrm{\&sigma;}}^2}=0.0014.$

2. will

Among the Reliability Function R of substitution step 2 (t), can obtain the fiduciary level value of product at moment t.

If the luminous power of fixed output quota product degenerates to initial value half, i.e. l=0.5y ₀The time, product failure.Will

The value substitution Reliability Function of l

$R\left(t\right)=\mathrm{\&Phi;}\left[\frac{l-y_0-d\left(s\right)t}{\mathrm{\&sigma;}\sqrt{t}}\right]-\exp \left(\frac{2d\left(s\right)(l-y_0)}{{\mathrm{\&sigma;}}^2}\right)\mathrm{\&Phi;}[-\frac{l-{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="H2009102429877E00021.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+d\left(s\right)t}{\mathrm{\&sigma;}\sqrt{t}}]$

Thereby obtaining product is that 0.9383,10 ten thousand hour fiduciary level is 0.9994 in 200,000 hours fiduciary levels of 25 ℃ of work.Fig. 2 is the curve map that is changed along with the time by the resulting production reliability of Reliability Function.

Claims (3)

1. based on the Bayesian reliability comprehensive estimation method of multisource degraded data, it is characterized in that, comprise following step:

The collection of step 1, multisource degraded data;

(1) definite degradation parameter that needs collection;

(2) collect product degradation information relevant in development stage, production phase, operational phase with fixed degradation parameter;

(3) degradation information of collecting is screened, put in order, obtain needed degraded data;

Degraded data should meet the following conditions;

1. choose the degradation information of same degradation parameter;

2. the degradation information of obtaining is a degraded data, or degradation information can be converted into degraded data;

For form is the degradation information of degraded data, directly uses; If degradation information known certain function or the function expression that be degraded data is converted into degraded data or obtains degraded data by method of emulation by known function;

3. the degradation mechanism of the degraded data of separate sources is identical;

(4) according to the source of degraded data, these degraded datas are divided, come from m information source, so just degraded data is divided into m group information source data as data;

Step 2, set up the degeneration statistical model and determine Reliability Function;

Specifically may further comprise the steps:

(1) determines the degradation model of product degradation parameter;

Use the method that returns that the time dependent degradation model of product degradation parameter is carried out the match modeling; When specifically the product degradation process being carried out the match modeling, set up the time dependent degradation model of the degradation parameter with distribution characteristics by following three kinds of methods as required:

1). suppose that the parameter in the time dependent definite function model of degradation parameter obeys certain distribution;

2). to the time dependent definite function of degradation parameter, add one and do not change the margin of error of obeying certain distribution in time;

3). to the time dependent definite function of degradation parameter, add stochastic process and describe;

(2) set up degeneration statistical model and definite Reliability Function;

Different situations when setting up degradation model are carried out conversion to degradation model, and degraded data are handled, and obtain applying to distribution and data mode in the bayes method; And, set up the degeneration statistical model of product and determine Reliability Function based on this in conjunction with inefficacy thresholding l and failure criteria;

Three kinds of modeling methods at described in the step 1 are specially:

1) supposes that the parameter in the time dependent definite function model of degradation parameter obeys certain distribution;

Certain distribution by parameter obedience in the function, other parameter in the function and variable are brought in this distribution, thereby set up a new distribution, and the degraded data behind the variation is used as new data, set up the degeneration statistical model, obtained Reliability Function R (t);

2), add one and do not change the margin of error of obeying certain distribution in time to the time dependent definite function of degradation parameter;

By certain distribution that the margin of error in the function is obeyed, parameter in the function and variable are brought in this distribution, thereby set up a new distribution; And the degraded data behind the variation used as new data, set up the degeneration statistical model, obtain Reliability Function R (t);

3), add stochastic process and describe to the time dependent definite function of degradation parameter;

For adding the time dependent definite function of performance degradation parameter that stochastic process is described, according to the character of stochastic process function is handled, set up the degeneration statistical model, obtain Reliability Function R (t);

When being broad sense Brownian Motion with Drift model:

Y(t)＝σB(t)+g(t，s)·t+y ₀????????????????????(1)

Wherein Y (t) is the degenerative process of product parameters; B (t) is 0 for average, variance be time t standard Brownian movement B (t) N (0, t); σ is a coefficient of diffusion, does not change with stress and time, is constant; (t s) is coefficient of deviation to g; y ₀Initial value for properties of product;

Therefore by the character of Brownian Motion with Drift as can be known, it is that (variance is σ to g for t, s) Δ t that the degeneration increment Delta Y of unit interval Δ t obeys average ²The normal distribution of Δ t, promptly

ΔY□N(g(t，s)·Δt，σ ²Δt)??????????????????(2)

As new data mode, formula (2) distributes as it with Δ Y, thereby has determined the form of distribution and data, has set up the degeneration statistical model;

If establish the failure threshold that l is a parameter, promptly establish Y (t)-l＜0 o'clock product failure; Utilize the Brownian Motion with Drift Reliability Model that Reliability Function is found the solution:

$R\left(t\right)=\mathrm{\&Phi;}\left[\frac{l-y_0-g(t,s)t}{\mathrm{\&sigma;}\sqrt{t}}\right]-\exp \left(\frac{2g(t,s)(l-y_0)}{{\mathrm{\&sigma;}}^2}\right)\mathrm{\&Phi;}\left[\frac{l-y_0+g(t,s)t}{\mathrm{\&sigma;}\sqrt{t}}\right]---\left(3\right)$

Step 3, degraded data pre-service;

Specifically may further comprise the steps:

(1) resulting degraded data is converted into new data mode;

According to distribution that obtains in the step 2 and data mode, degraded data is carried out conversion, make it possess statistical property and obey this distribution;

(2) determine that stress amounts to model;

Each stress of product degradation process all has acceleration model or reduced factor separately, according to degradation parameter and stress, determines that stress amounts to acceleration model or the reduced factor that model is a stress;

(3) degraded data under the different stress levels is folded under the same stress level;

Be specially:

1. if it is acceleration model G (s) that the middle stress of above-mentioned steps (2) is amounted to model,, uses the method match of regression fit and also set up a certain eigenwert z relevant in the degenerative process with the time at first according to the degraded data under the different stress level s _sAnd the relational expression between the stress level s:

z _s＝G(s)?????????????????(4)

Obtain the value of a certain relevant eigenwert zs under the different stress level s with the time by formula (4);

Secondly, with stress s _iDegraded data yi under the level is folded to required stress level s _iUnder degraded data y _i, if can set up degraded data y and a certain eigenwert z relevant with the time _sRelation:

y＝f(z _s)?????????????????(5)

Wherein, formula (5) is by degradation model and a certain eigenwert z relevant with the time _sConcrete meaning determine; So as can be known by formula (4), (5):

$y_i=\frac{f\left(G\left(s_i\right)\right)}{f\left(G\left(s_j\right)\right)}\&CenterDot;y_j---\left(6\right)$

Thereby finished amounting between the degraded data under the different stress levels;

2. be reduced factor k if the middle stress of above-mentioned steps (2) is amounted to model _Ij, by the reduced factor k between given stress level _IjWith stress level s _iUnder a certain eigenwert z _SiBe folded to stress level s _jUnder a certain eigenwert z _Sj:

z _si＝k _ij·z _sj????????????(7)

Finished amounting to of degraded data between different stress levels by formula (5) then;

(4) form of each information source data is unified;

If can not set up described degraded data y of formula (5) and a certain eigenwert z relevant in the step (3) with the time _sRelation, then utilize the time interval Δ t between each degraded data to set up it and a certain eigenwert z relevant with the time _sRelation, then have:

Δt＝f(z _s)???????????????(8)

Obtain by formula (4), (8):

$\mathrm{\&Delta;}{t}_i=\frac{f\left(G\left(s_i\right)\right)}{f\left(G\left(s_j\right)\right)}\&CenterDot;\mathrm{\&Delta;}{t}_j---\left(9\right)$

Along with the change of the time interval Δ t between degraded data, each degraded data has been represented the parameter degradation values in the different time, need unify the degraded data form, and concrete grammar is:

At first, obtain each degraded data time interval Δ t by formula (9) _i, obtain their lowest common multiple Δ t _G, obtain Δ t _GWith Δ t _iMultiple concern p _i=Δ t _G/ Δ t _i

Then, with every p of each degraded data _iIndividual data point merges, with p _iThe amount of degradation of individual degraded data merges to a point, and then the time interval of all degraded datas all equates, the degraded data unity of form;

(5) consistency check;

Each information source data that step (4) is obtained as the prior imformation in the bayes method, will speed up in the degradation experiment degraded data through step (4) as the sample information in the bayes method, and the distribution that step 2 is obtained is as population distribution;

Whether adopt rank test method check prior imformation compatible with sample information, concrete grammar is:

If $Y_i=(Y_{i1},\&CenterDot;\&CenterDot;\&CenterDot;,Y_{i{n}_1})$ Be i group prior imformation, $X=(X_1,\&CenterDot;\&CenterDot;\&CenterDot;,X_{n_2})$ Be sample information, introduce and compete to select hypothesis mutually:

H ₀: X and Y _iBelong to same overall; H ₁: X and Y _iDo not belong to same overall;

If X _iWith Y _iBy check, think that two increments do not belong to same overall; Otherwise it is same overall that two increments belong to, and this group prior imformation is removed;

What step 4, prior distribution, posteriority distributed determines and fusion;

(1) according to prior imformation, population distribution, the actual conditions of sample information, definite method of selection prior distribution;

(2) determine the prior distribution of each information source;

For the m group prior imformation of m information source, each group prior imformation uses identical prior distribution to determine method, obtains prior distribution π separately _i(θ), and π _i(θ) obey identical prior distribution family;

(3) robustness check;

Adopt the robustness method of inspection to judge whether determined prior distribution meets the requirements;

If known prior distribution π _i(θ), population distribution f (x| θ) and sample information x, the formula of Marginal density function, is so:

$m_i\left(x\right)=\underset{\mathrm{\&Theta;}}{\&Integral;}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;}---\left(10\right)$

By the m that tries to achieve _i(x) value judges whether determined prior distribution is sane, sets an acceptable value cv as required, if m _i(x)＜cv then needs to use other method to redefine prior distribution;

(4),, determine that the posteriority of each information source distributes in conjunction with the prior distribution of each information source by sample information;

By prior distribution π _i(θ), population distribution f (x| θ) and sample information x obtain the posteriority distribution π of θ _i(θ | x), the posteriority distribution formula is:

${\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right|x)=\frac{{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})}{\underset{\mathrm{\&Theta;}}{\&Integral;}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;}}---\left(11\right)$

(5) posteriority of finding the solution and obtaining after each information source merges by weighting coefficient distributes;

Be specially:

Known population distribution f (x| θ) obtains prior distribution π respectively by m information source ₁(θ), π ₂(θ) ... π _m(θ), the prior distribution after merging so is:

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)---\left(12\right)$

ε wherein _iBe weighting coefficient:

$\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i=1---\left(13\right)$

Adopt correlation coefficient process to find the solution ε ⁱ

This moment, posteriority was distributed as:

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right|x)=\frac{\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})}{\underset{\mathrm{\&Theta;}}{\&Integral;}\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;}}---\left(14\right)$

So the posteriority after merging is distributed as

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right|x)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&epsiv;}}_{i}m\left(x\right|{\mathrm{\&pi;}}_i)}{m\left(x\right|\mathrm{\&pi;})}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right|x)---\left(15\right)$

Wherein, $m\left(x\right|{\mathrm{\&pi;}}_i)=\underset{\mathrm{\&Theta;}}{\&Integral;}{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right)f\left(x\right|\mathrm{\&theta;})\mathrm{d\&theta;},$ M (x| π)=∑ m (x| π _i); Order ${\mathrm{\&xi;}}_i=\frac{{\mathrm{\&epsiv;}}_{i}m\left(x\right|{\mathrm{\&pi;}}_i)}{m\left(x\right|\mathrm{\&pi;})},$ So

$\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i=1---\left(16\right)$

Be so

$\mathrm{\&pi;}\left(\mathrm{\&theta;}\right|x)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i{\mathrm{\&pi;}}_i\left(\mathrm{\&theta;}\right|x)---\left(17\right)$

Secondly, to π _i(θ) get expectation u _i, to π _i(θ | x) get expectation u ' _i, variances sigma ⁱ ₂, to π (θ | x) get expectation $u^{\&prime;}=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i{u}_i^{\&prime;};$ By the emulation sampling, obtain m+1 group u _i, u ' _i, u ' estimated value, and σ ₁ ², σ ₂ ²Estimated value; Bring data into system of equations:

$\left\{\begin{array}{c}{u}_i^{\&prime;}=b_{0i}^{\&prime;}+b_{1i}^{\&prime;}{u}_1+b_{2i}^{\&prime;}{u}_2+\&CenterDot;\&CenterDot;\&CenterDot;b_{\mathrm{mi}}^{\&prime;}{u}_m\\ u^{\&prime;}=b_0+b_1{u}_1+b_2{u}_2+\&CenterDot;\&CenterDot;\&CenterDot;b_m{u}_m\\ r_i=\frac{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{b}_{\mathrm{ji}}^{\&prime;}{b}_j{\mathrm{\&sigma;}}_j^2}{\sqrt{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{b_{\mathrm{ji}}^{\&prime;}}^2{\mathrm{\&sigma;}}_j^2}\sqrt{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{b_j^2\mathrm{\&sigma;}}_j^2}}\\ {\mathrm{\&epsiv;}}_i=r_i/\left(\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{r}_i\right)\end{array}\right.$

Obtain ε thereby find the solution _iAnd ξ _iValue; And then the posteriority distribution π of formula (17) after obtaining merging (θ | x);

Step 5, assessment reliability of products;

(1) by the posteriority distribution π after merging (θ | x),, obtain the assessed value of the parameter θ of population distribution in conjunction with the Bayesian Estimation method

Wherein:

$\widehat{\mathrm{\&theta;}}E\left(\mathrm{\&theta;}\right|x)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i{E}_i\left(\mathrm{\&theta;}\right|x)---\left(18\right)$

(2) will

Among the Reliability Function R of substitution step 2 (t), obtain the fiduciary level value of product at moment t; Ask constantly the interval of the reliability R (t) of t to estimate as needs, by posteriority distribution π (θ | x) interval that obtains parameter θ is estimated, estimates in the interval of the reliability R (t) of moment t thereby obtain product.

2. the Bayesian reliability comprehensive estimation method based on multisource degraded data according to claim 1, it is characterized in that, in (2) of step 2, when the parameter in the time dependent definite function model of hypothesis degradation parameter is obeyed certain distribution, when the time dependent definite function model of certain product degradation parameter is:

y(t)＝β ₀+β ₁t??????????????(19)

Wherein y (t) is the product degradation parameter, β ₀, β ₁Be parameter item, t is the time; If β ₁N (θ ₁, σ ₁ ²), the character according to normal distribution obtains:

y(t)□N(β ₀+θ ₁t，σ ₁ ²t ²)???(20)

And then obtain:

Will

As new data mode, formula (3) distributes as it, thereby has determined the form of distribution and data, has set up the degeneration statistical model;

If establish the failure threshold that l is a parameter, promptly establish y (t)-l＜0 o'clock product failure, obtain the Reliability Function of product by the formula of deriving above:

$R\left(t\right)=P(y\left(t\right)-l>0)=\mathrm{\&Phi;}\left(\frac{{\mathrm{\&beta;}}_0+{\mathrm{\&theta;}}_{1}t-l}{\sqrt{{\mathrm{\&sigma;}}_1^2{t}^2}}\right)---\left(22\right)$

。

3. the Bayesian reliability comprehensive estimation method based on multisource degraded data according to claim 1, it is characterized in that, in (2) of step 2, to the time dependent definite function of degradation parameter, add when one the margin of error of certain distribution is obeyed in variation in time, suppose that the time dependent definite function model of certain product degradation parameter is:

y(t)＝Ct+ε???????????????????(23)

Wherein y (t) is the product degradation parameter, and C is a constant, ε N (θ, σ ²), t is the time; Then can obtain according to the character of normal distribution

y(t)□N(Ct+θ，σ ²)???????????(24)

And then obtain

y(Δt)□N(CΔt，0)????????????(25)

As new data mode, formula (7) distributes as it with y (Δ t), thereby has determined the form of distribution and data, has set up the degeneration statistical model;

If establish the failure threshold that l is a parameter, promptly establish y (t)-l＜0 o'clock product failure, so just obtain the Reliability Function of product by the formula of deriving above:

$R\left(t\right)=P(y\left(t\right)-l>0)=\mathrm{\&Phi;}\left(\frac{\mathrm{Ct}+\mathrm{\&theta;}-l}{\sqrt{{\mathrm{\&sigma;}}^2}}\right)---\left(26\right)$

。

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